2003 AACTM Lewis-Parker Lecturer
Prof. Olav Kallenberg
Auburn University

Lewis-Parker Lecture
Title: The Basic Symmetries of Probability Theory
Abstract: We consider random sequences or processes whose distributions are invariant under certain groups (or semigroups) of transformations. Three such symmetries---invariance under contractions, permutations, and rotations---turn out to be especially fundamental, and their study leads to a rich theory that relates to practically every other branch of modern probability theory. It is impossible in a short talk to do justice to such a vast area of research. (The speaker is currently writing a monograph on the theory that will easily run to some 500 pages.) Our present aim is only to discuss, in broad and elementary terms, some basic connections between the various symmetries, some relations to classical analysis, and some surprising applications to gambling and sampling. We hope to conclude with a brief sketch of the more advanced multivariate theory.